SUBCONVEXITY · New bounds for automorphic L-functions
6РП — Действия „Мария Кюри“
- Период
- 2006-11-01 → 2008-10-31
- Финансиране от ЕС
- 108 988 €
- Участници
- 1
- Схема
- EIF
Линиите свързват координатора с партньорите.
Накратко на български
Автоморфните L-функции, които са разширение на синуса и косинуса, се изследват чрез търсене на нови граници за техните стойности. Това помага за по-доброто разбиране на разпределението на числата и решенията на квадратни уравнения с цели числа.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Final Activity Report Summary - SUBCONVEXITY (New bounds for automorphic L-functions)
Automorphic forms are symmetric waves, generalisations of the well-known sine and cosine functions. The sine and cosine functions are defined on the line of real numbers, and their periodicity can be described as invariance under certain translations of the line. Automorphic forms are defined on spaces with rich geometry, where more complicated symmetries are present. They are the harmonic components of symmetric functions on the space; similarly as nice periodic functions on the line can be decomposed into sines and cosines. Mysteriously and magically, automorphic forms can make very deep properties of the integers visible. L-functions are useful in formulating, conjecturing and in some cases proving such properties. In short, automorphic L-functions provide a certain key to understanding the integers. A famous unproved property of automorphic L-functions concerns the distribution of their zeros. Namely, it is expected that all nontrivial zeros of L-functions are located on a certain line of the plane of complex numbers, the axis of symmetry of the L-function. This property is the Riemann Hypothesis, one of the most important unsolved problems in mathematics. It has a number of deep and interesting consequences, a notable one being the Lindelöf Hypothesis stating that automorphic L-functions are not too large on their axis of symmetry. The objective of the project was to make progress towards this weaker hypothesis, precisely to exhibit new bounds for automorphic L-functions. Several new bounds have been found, and a new technique has been developed which has the potential of treating cases where previous approaches were unsuccessful. Among the consequences are a better understanding of solutions of quadratic equations in three integral variables and with integral coefficients. Geometrically speaking, this corresponds to a better understanding of lattice points on ellipsoids in three-dimensional space.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
Number theory is among the oldest and most central disciplines in mathematics. It has truly fascinating connections to algebraic geometry, combinatorics, ergodic theory, representation theory, and mathematical physics. Many of these connections as well as deep intrinsic properties are formulated in the language of L-functions.An important task is to establish subconvex bounds for automorphic L-functions. Such bounds reflect the arithmetic nature of their source objects as they cannot be derived from simple analytic principles. In return, they provide the key to the solution of several difficult Diophantine problems addressing equidistribution phenomena.Proving these bounds unconditionally also sheds light on the Grand Riemann Hypothesis as they are consequences of it. Gergely Harcos, the researcher of the present proposal, is an active participant in this area. After a decade of successful work in the United States he intends to return to Europe and pursue his research program in the stimulating and supportive environment of the Renyi Institute.His efforts would be enforced by the complementary skills of the experts at the host in the theory of algebraic groups and automorphic forms, and in analytic number theory. In addition, the researcher would receive advanced training in combinatorial number theory and prime number theory, which are important for his future career.The ultimate goal for the researcher is to integrate his research at the host and start new or revive past scientific collaboration with members of the host. The project would strengthen and diversify the mathematical profile of Hungary, one of the new Member States of the European Union, by introducing an important new line of mainstream research and by providing new links to colleagues worldwide.The project would also contribute towards reversing brain drain. In short, the proposed project would enhance the potential and the attractiveness of the European Research Area.
Оригинален текст от CORDIS (на английски).
Участници
- ALFRED RENYI INSTITUTE OF MATHEMATICS - HUNGARIAN ACADEMY OF SCIENCES · BUDAPESTКоординаторУнгария
Връзки
Данни: CORDIS, © Европейски съюз